Plot function with infimum over specific domain

Question

I'm fairly new to Mathematica. Im trying to plot a function but i cannot seem to get it right.

The function is: $f:\mathbb{R} \to \mathbb{R},f(x)=inf\{\vert x-k\vert \vert k\in \mathbb{Z}\}$

I tried defining it seperatly using the Min[] function for the infimum and Abs[] for the absolute value. I read about using something called ImplicitRegion[] but i do not understand how i would use it such that i can tell that $z \in \mathbb{Z}$.

It would be much appreciated if you could tell me how i can plot such a function.

Thank you for helping me out.

Answer 1

Edit

I think you want to plot the function which only at countable number of points are non-differentiable. Such function is a periodic function.

We can deduce this result all by Mathematica.

Simplify[Min[Abs[x - β], Abs[x - (i - 1)], Abs[x - i], 
  Abs[x - (i + 1)], 
  Abs[x - α]], {i <= x <= i + 1, α >= i + 1, β <= 
   i - 1}]

Min[1 + i - x, -i + x]

It means that for $i\leq x \leq i+1$, $$\inf_\limits{i\in \mathbb{Z}}| x-i|=\min\{1+i+x,-i+x\}$$

So we can define the function If[i <= x <= i + 1, Min[1 + i - x, -i + x]] for a fixed i

Manipulate[
 Plot[If[i <= x <= i + 1, Min[1 + i - x, -i + x]], {x, -6, 6}, 
  AspectRatio -> Automatic, Axes -> {True, False}], {i, -5, 5, 1}]

Show[Table[
  Plot[If[i <= x <= i + 1, Min[1 + i - x, -i + x]], {x, -6, 6}, 
   AspectRatio -> Automatic, Axes -> {True, False}], {i, -6, 5, 1}]]

Original

Manipulate[
 Plot[Min[Table[Abs[x - i], {i, -n, n}]], {x, -n, n}, 
  AspectRatio -> Automatic], {n, 1, 10, 1}]